Moment Arm and Joint Angle

Moment (lever) arm of a muscle acting on a joint changes with the angle between the two bones articulating in that joint. Consider, for example, the moment arms of the forearm flexors, biceps, and the brachioradialis muscles shown in Fig. 6.7a. The moment arm of biceps is nearly equal to zero when the angle between the upper arm and the forearm is 180°. The moment arm increases as the joint angle 6 decreases from 180° toward 90°. What is the relation between the moment arm and the joint angle? This question can be addressed using geometric relationships. We illustrate the analysis by considering the dependence of the lever arm of biceps brachia on the joint angle. Figure 6.7b illustrates this muscle as a cord spanning through the humerus and inserting at the radius. Actually, the geometry chosen is a simplification of the topology of the biceps. This is a biarticular muscle in which both heads of the biceps arise at the scapula rather than at the proximal head of humerus. Additionally, a strong membranous band arising from the tendon of biceps attaches to the ulna. Nevertheless, Fig. 6.7b is useful in investigating the action of this muscle group on the elbow joint.

Let a denote the length of the humerus, b denote the distance between the center of rotation of the elbow joint and the point of insertion of biceps on radius, and 6 be the angle between the humerus and radius (Fig. 6.7b). In this figure the length of the biceps muscle-tendon cord is represented by the letter c and the normal distance from the center of line of application of the biceps force to the center of rotation of the elbow joint by the letter d. Note that d is the moment arm of the biceps force with respect to the elbow joint. Using the cosine law, one can express the muscle-tendon length c as a function of a, b, and 6:

Because the tension in the muscle-tendon complex depends on the length of the complex, it is important to be able to compute this length as a function of the joint angle 6.

Figure 6.7a-c. Flexion of the forearm as a result of contraction of biceps brachii and brachioradialis (a). The symbols S and E identify the centers of rotation of the shoulder and elbow muscle, respectively. The biceps muscle group is represented by a cord joining points S to B, and brachioradialis by a cord joining points A and C. The length parameters a, b, c, d, and angles 6 and 6\ used in the analysis are identified in (b). Note that a and b refer to lengths along the adjoining bones whereas c is the length of the muscle-tendon complex. The moment arm is shown by the letter d in the figure. Forces acting on the forearm during flexion against a resistance of m = 10 kg are shown in (c). Weight of the forearm is neglected in the free-body diagram.

Figure 6.7a-c. Flexion of the forearm as a result of contraction of biceps brachii and brachioradialis (a). The symbols S and E identify the centers of rotation of the shoulder and elbow muscle, respectively. The biceps muscle group is represented by a cord joining points S to B, and brachioradialis by a cord joining points A and C. The length parameters a, b, c, d, and angles 6 and 6\ used in the analysis are identified in (b). Note that a and b refer to lengths along the adjoining bones whereas c is the length of the muscle-tendon complex. The moment arm is shown by the letter d in the figure. Forces acting on the forearm during flexion against a resistance of m = 10 kg are shown in (c). Weight of the forearm is neglected in the free-body diagram.

Computation of d from the anthropometric data is also straightforward. Figure 6.7b indicates that d = a sin 9\ (6.11)

in which 61 denotes the angle between the long axis of humerus and line of application of the biceps force. Using the cosine law, this angle can be expressed as a function of a, b, and c:

cos 61 = (-b2 + a2 + c2)/2ac sin 61 = (1 - cos2 61)1/2

Equations 6.10 and 6.12 can be used to determine the muscle-tendon length and the moment arm as a function of the joint angle.

Example 6.6. Moment Arm of Biceps. Consider a person flexing the forearm against a 10-kg weight as shown in Fig. 6.7c. Assume that the arm is weightless and that the biceps muscle is the one muscle involved in the flexion. The length a of the humerus is 32 cm and distance b between the point of insertion of biceps into radius and the elbow joint is 9 cm. The length of the forearm (including half the length of the hand) is 34 cm. Determine the moment arm of the biceps, and the biceps force at joint angle 6 = 170°, 135°, 90°, and 45°.

Solution: Using Eqn. 6.10 and the parameter values given for a and b, we compute the length c of the muscle-tendon complex as a function of joint angle 6. Then, we use Eqn. 6.12 to determine the moment arm d:

Next, we compute the moment exerted by the biceps force about the elbow joint. Because the motion occurs slowly, we can neglect the iner-tial effects and set the sum of moments acting on the forearm at the elbow equal to zero.

where Mb is the counterclockwise moment created by the biceps force at the elbow, m = 10 kg is the weight to be lifted, and Lf = 34 cm is the length of the forearm (Fig. 6.7c). The moment Mb can be expressed as follows:

in which d is the moment arm and Fb is the magnitude of the biceps force. Using these two equations along with Eqns. 6.10 and 6.12, we compute the moment and the force produced by the biceps muscle as a function of the joint angle:

If the moment arm were assumed constant throughout the range of 6, the biceps would have been predicted to produce the highest force at 6 = 90°. Obviously, this is not the case. A word of caution here: not all muscles of the upper and lower limbs experience as great a variation of the moment arm with the joint angle as does the biceps. For example, the moment arm of the triceps is much less dependent on the joint angle in comparison with biceps (not shown).

The use of dumbbells gives you a much more comprehensive strengthening effect because the workout engages your stabilizer muscles, in addition to the muscle you may be pin-pointing. Without all of the belts and artificial stabilizers of a machine, you also engage your core muscles, which are your body's natural stabilizers.